Vibrational Polaritons in Disordered Molecular Ensembles

Disorder is an intrinsic attribute of any realistic molecular system. It is known to lead to localization, which hampers efficient transport. It was recently proposed that in molecular ensembles strongly coupled to photonic cavities, moderate disorder leads to delocalization and increases of the transport and chemical reaction rates. Vibrational polaritons involve molecular vibrations hybridized with an infrared cavity. When the coupling strength largely exceeds the molecular inhomogeneity, polaritons are unaffected by disorder. However, in many experiments, such a homogeneous limit does not apply. We investigated vibrational polaritons involving molecular ensembles with systematically modified disorder. Counterintuitively, moderate disorder leads to an increase in Rabi splitting and the modification of the polariton bandwidths. Experimental spectroscopic data agree with a Tavis–Cummings-like model that suggests enhanced delocalization of the reservoir states occurs via the admixture of the cavity mode. Our results provide new insights into the paradigm of disorder-induced cavity-assisted delocalization in molecular polaritons.

M olecular polaritons are hybrid light−matter states formed when an ensemble of N molecular transitions strongly interacts with the "privileged" electromagnetic mode of an optical cavity. 1,2 Polaritons are envisioned to play revolutionary roles in emerging applications ranging from quantum technologies to chemical catalysis; 3−8 thus, they have been investigated extensively. 9−11 Experimental results are frequently analyzed within the framework of cavity quantum electrodynamics. Specifically, the Tavis−Cummings (TC) model, 12,13 which describes the interaction between a cavity and an ensemble of degenerate molecular modes, predicts that a manifold of the reservoir states, potentially with new and important properties, is formed in addition to a pair of polariton states. 14−17 Recently, the TC model was generalized to account for molecular disorder. 18−23 Disorder is an intrinsic property of any realistic molecular system, which one expects to affect the composition and dynamics of both polaritons and reservoir states. [18][19][20]24,25 When the molecule−cavity coupling strength largely exceeds the molecular inhomogeneous bandwidth, the effect of the disorder can be neglected. 26 However, when the ratio between the two is smaller than about an order of magnitude, which is a frequent situation in polaritonic systems involving molecular ensembles, 27−29 the conclusions drawn, based on the application of homogeneous models, might not rigorously hold. Recent studies theorize that reservoir states delocalize over multiple molecules in the presence of disorder, thus facilitating efficient transport 18,25,30 and vacuum-field catalysis, 20,31 which is strikingly opposed to the typical effect that disorder has in the absence of coupling to a photonic cavity. 32 However, in most practical realizations of these systems, experimentally controlling the molecular disorder is formidable, and no direct experimental demonstration of the associated effects has been reported.
Consider the first excitation tier of the system schematically illustrated in Figure 1a, where an ensemble of noninteracting two-level systems m i ( = i N 1, ..., ), representing molecular transitions, is coupled with rates g i to a cavity mode c. The corresponding TC-like Hamiltonian [18][19][20][21][22]25 where † a i and † b are the molecular and photonic creation operators. To account for intrinsic losses, c and m i , are assumed to be complex, = E i j j j , where j is the dissipation rate associated with the homogeneous bandwidth of the corresponding transition. 33 The values E m i , are normally distributed about the mean E m with the standard deviation (inhomogeneous bandwidth) m . For uniform values = g g i , a transformation can be made to the so-called bright−dark states basis (Figure 1a), 11,13,18,19 where the molecular manifold is transformed into a many-body superoscillator bright state | = | B e , whereas |d k are coupled to |B with rates k .
The eigenstates of the Hamiltonian (eq 1) involve the upper (UP) and the lower (LP) polariton states and the reservoir states |R k . When g c tot , Γ m , σ m , and | | E E c m , the effect of k can be neglected, and the system follows the homogeneous limit. 26 In such a regime, | | R d k k are often referred to as "dark" because their transition dipoles vanish. 17,34 Since k is negligible, |R k do not involve any cavity component, lack dispersion, and are localized. Furthermore, in this regime, polaritons appear to be insensitive to the molecular disorder and feature a homogeneous line shape whose bandwidth is given by In contrast, when a homogeneous regime does not apply, the disorder can be viewed as a perturbation that facilitates admixing of the cavity mode into reservoir via the coupling constants k , which lifts the reservoir's dark character and delocalizes the quantum states. We refer to this as an inhomogeneous regime.
In the present work, we examined the role of disorder on polaritons formed by the carbonyl stretching (CO) vibrational modes of W(CO) 6 molecules in solution, which were coupled to an open cavity. Our experimental system allows for the independent systematic control of both the molecular inhomogeneous bandwidth and its collective coupling strength to the cavity. The former is achieved via controlling solute− solvent interactions by changing the solvent polarity, 35,36 whereas the latter is achieved via controlling the concentration of solute molecules. 37 We demonstrate that, counterintuitively, in the inhomogeneous regime, an increase in molecular disorder leads to an increase in polariton splitting and line width modification, as schematically illustrated in Figure 1b. With the help of the theoretical analysis, we further show that both observations arise from the disorder-induced admixture of the cavity mode into the reservoir manifold. We quantify this effect with the effective coupling constant ( ) m eff , which describes the coupling between the bright and dark states, and we demonstrate its correlation with the degree of delocalization of the reservoir states.
Experimentally, an open cavity was constructed with a highquality antenna-lattice resonance (ALR) of the half-wavelength infrared antenna array, 38,39 where individual dipolar antenna resonances couple to the in-plane lattice diffraction order. 40,41 Unlike the commonly used Fabry−Perot-style optical cavities, in our experiments, the optical extinction occurs only at the ALR frequency while the nearby spectral region remains transparent, thus allowing direct spectroscopic access both to the polaritons and to the reservoir transitions. 38    and has a quality factor of Q ∼ 115. The choice of the two fully miscible solvents with similar refractive indices but with different polarities allows us to efficiently control the molecular disorder without modifying the frequency or bandwidth of the ALR.
The molecular inhomogeneous bandwidth was systematically varied by preparing different mixtures of n-octane and 1chlorobutane. As shown in Figure 2b, when the fraction of 1chlorobutane increases, the CO transition broadens and redshifts from m oct = 1982 cm −1 to m 1chl = 1977 cm −1 (vibrational Stark effect 35,36 ). The broadening reflects increases in both the homogeneous and inhomogeneous line shape components with the increasing solvent polarity. The increase in inhomogeneity indicates the presence of multiple solvation shell configurations created by 1-chlorobutane around the CO group. The spectra fit well with the Voigt line shape, which was previously shown to describe the dynamics of CO transition in W(CO) 6 in aliphatic solvents. 45,46 However, because fitting the linear spectrum to the Voigt profile may produce homogeneous and inhomogeneous widths with a high covariance, we determined their values for each solvent mixture using 2D IR spectroscopy, as described in details in the Supporting Information. The results are shown in Figures S1 and S2, and all the obtained parameters are summarized in Table S1.
Changing the solvent composition from n-octane to 1chlorobutane leads to an increase in the inhomogeneous bandwidth from m The antenna array was covered by a 3.6 μmthick 20 mM solution of W(CO) 6 and capped with an additional CaF 2 window. The dispersion of the vibrational polaritons measured for different incident angles, shown in Figure 2c, features an avoided crossing between the ALR and the CO modes at around°12 in neat n-octane and that at°11 in neat 1chlorobutane. To compensate for the solvent-induced shift of the CO mode frequency and to maintain the resonant conditions, the ALR transition was tuned without affecting the ALR quality by slightly changing the angle of the TMpolarized incident light 39 with a precision goniometer, as shown in Figure S3 of the Supporting Information.
The background-subtracted spectra of the strongly coupled system are shown in Figure 2d for the two neat solvents. The background spectra were collected for each sample away from the array. Background subtraction allows for both the efficient removal of the contribution from the uncoupled W(CO) 6 molecules 47,48 and the detailed resolution of the polariton line shapes (see also Figure S4 of the Supporting Information). Polariton spectra were fit to two Lorentzian profiles, and the vacuum Rabi splitting = E E UP LP was calculated from the peaks' central frequencies obtained for the solvent mixtures of different compositions, as plotted in Figure 3a. To keep the coupling strength g tot constant, the concentration was kept at 20 mM in all samples. Using Beer's law, we verified that the extinction coefficient of W(CO) 6 does not differ significantly in these solvents, as demonstrated in Figure S5 of the Supporting Information.
Remarkably, changing the solvent polarity leads to increase in by up to ∼35%. While both m and m affect the value of (see eq 2 below), m accounts only for a small fraction of the observed . As shown in Figure 3a, the maximum splitting was obtained for m ≈ 3.5 cm −1 , which was followed by a slight decrease. In neat solvents, the Rabi splittings were oct = 22.8 cm −1 and 1chl = 28.5 cm −1 . These splitting values exceed the bandwidth of the uncoupled transitions of both the ALR ( ALR oct = ALR 1chl = 18 cm −1 , fwhm) and the total bandwidth of the CO mode (fwhm m oct = 5.1 cm −1 and fwhm m 1chl = 13.7 cm −1 ) by less than 10-fold, indicating the inhomogeneous regime, as discussed above. The increase in the Rabi frequency is qualitatively reproduced by the real parts of the eigenvalues of the Hamiltonian (eq 1), 49 accounting for the experimentally determined m , m , and g tot values, as shown in Figure 3b. The numerical model predicted values similar to the experimental ones, but the maximal splitting was obtained for m = 9 cm −1 , which was larger than that in the experiment. To facilitate an intuitive interpretation of the results in Figure 3, we reduced the bright−dark state representation of the Hamiltonian in eq 1 to an effective three-state model, which is schematically illustrated in Figure 4a. Here, the cavity mode is coupled to the bright state |B by g tot as described earlier, whereas the dark states are represented by a single surrogate collective state |D coupled to |B with the effective rate eff , where the values depend on the disorder = ( ) m eff eff . 19 The new Hamiltonian is = i k j j j j j j j j j j j j j y { z z z z z z z z z z z z z Because the collective |B and |D states are purely molecular, the energies of these states are E B = E D = E m (at the resonance, E c is also equal to E m ), the corresponding decay rates are B = m , and D can be chosen to effectively represent the bandwidth of the molecular transition.
An approximate analytical solution for the complex eigenvalues, namely, = + i r i , can be obtained if we further assume that D m . Although such an approximation is certainly not valid for analyzing the reservoir modes, we find it useful for analyzing polaritons. 19 Under these approximations, the real part of the eigenvalues is  Figure 4b. For low-to-moderate disorders, eff grows, reaching the maximal value at m ≈ 9 cm −1 , where is the largest. For larger disorder, eff decreases, leading to a decrease in . 19 Next, we analyze the polaritons' bandwidths. For the 20 mM solutions, bandwidths of UP oct = 9.4 cm −1 , LP oct = 10.6 cm −1 , 1 UP chl = 8.6 cm −1 , and LP 1chl = 11.2 cm −1 were obtained for neat n-octane and 1-chlorobutane. Interestingly, for all transitions, the bandwidths were below those expected in the homogeneous limit, where hom oct = 11.6 cm −1 and hom 1chl = 12.6 cm −1 .
To examine polariton line narrowing, we measured the change in the bandwidth for different g tot values, which were obtained with different concentrations of the W(CO) 6 molecules. As shown in Figure 4, as the coupling strength increases , LP decreases from the bare-cavity value c = 18 cm −1 (for g tot = 0) to Agreement between the experimental and theoretical results allowed us to use the latter to explore why there was an increase in Rabi splitting with an increase in the molecular disorder and the modification of polariton line widths with an increase in the coupling strength. To this end, we inspected the compositions of the corresponding eigenstates the Hopfield coefficients describing the admixture of the unperturbed state j into the eigenstate k, as shown in Figure 5. Generally, with the increase in inhomogeneity, the admixing of the cavity mode into polaritons decreases, whereas the admixing of the corresponding molecular component increases. In our experiments, < m c , which leads to narrowing  respectively, as shown in Figure 5. Analogously, for > m c , the broadening of polariton lines beyond the hom is expected. While naturally | | c R 2 i = 0 for m = 0, for m ≠ 0, an admixture of the cavity and the reservoir leads to the delocalization of the R i states, where the degree of delocalization depends on the interplay between m and g tot . The degree of delocalization is frequently quantified by the corresponding participation ratio (PR), which for the local states j delocalized over the eigenstate R i is given by 50,51 The average of PR over all the reservoir states in the homogeneous case is PR hom = 2.5. 20,25 For the coupling strength equivalent to that of the 20 nm W(CO) 6 solutions, with an increase in disorder, PR closely follows the trend of the change in eff with m , as shown in Figure 4b. Here, PR increases, reaching a maximal value of ca. 3 for m ≈ 9 cm −1 (maximal PR = 55, std = 2.3), then decreases toward the localized states when g tot is insufficient to overcome the disorder.
PR depends on the interplay between g tot and m . In the case of a weak disorder, when g tot increases, PR briefly increases, reaching the value of PR ≈ 3 near the exceptional point where polariton splitting emerges, 52 and then approaches PR hom (see Figure 5). As expected, the coupling strength required to surpass the exceptional point is larger in the case of stronger disorder; 21,53 however, the region of PR ≈ 3 is extended over a larger range of g tot values, compared with the case of weak disorder. Interestingly, strongly disordered ensembles involve fewer reservoir states delocalized over a large number of molecules compared to weakly disordered ensembles. This can be seen from the PR statistics in both cases (N = 40 000), which show that for PR = 3 the standard deviation of the PR values distribution is ∼8 for m = 3 cm −1 but only ∼3 for m = 8 cm −1 . The maximally delocalized states have PR values of 370 and 165.
Our calculations suggest that delocalization of reservoir modes is closely correlated with the admixture of the cavity mode (see Figure 5). For both cases of both weak and strong disorder, high PR values were observed when the Hopfield coefficients | | R c 2 i were high. Since in our experiments < m c , we also consistently found that < , UP LP hom , which supports the hypothesis that the delocalization of reservoir modes is enhanced due to the cavity admixture.
In conclusion, we demonstrated that disorder within the molecular ensemble, which strongly interacts with the optical cavity, affects the polariton spectrum, leading to a larger Rabi splitting and a modified polariton bandwidth. We also showed that these phenomena are correlated with the delocalization of the reservoir states, which is enhanced by the admixture of the photonic mode into reservoir. Our experimental results, supported by theoretical models, indicate that in the regime of a moderate disorder, coupling to the cavity can allow the well-known effect of localization to be overcome. Thus, our results provide new insights into the strong interaction between matter and vacuum fields, which can be harnessed and utilized in future quantum technologies.

■ ASSOCIATED CONTENT Data Availability Statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Linear and third-order nonlinear spectra of W(CO) 6 molecules, concentration dependence of linear signals, angle-tuning of the ALR resonances, raw and background-subtracted absorption spectra of the polariton states, and results of the Voigt line shape analysis of molecular transitions for all solvent mixtures used in the experiments (PDF)